Under virtually all circumstances, they seem to be so stupid and socially awkward that they likely have a mental disorder. But put them in the right kind of situation, and they'll demonstrate outstanding brilliance.

Mathematics (or a certain area of it) is a common area for Savant skills to show up. A few such people in real life are amazingly gifted at calendar-based calculations. Give them a specific date and they can tell you in seconds what day of the week it was or will be. Intuitive mathematical calculations, such as multiplication of large numbers or the identification of primes, also occur. Non-mathematical abilities include precocious musical ability, including perfect pitch and intuitive ability to play an instrument; unusual artistic ability, especially the ability to draw anything after having seen it once; prodigious feats of memory, such as memorizing entire books or a million digits of pi; or even the ability to learn a new language in a week.

When Hollywood uses Savant characters, the most frequent talent the character will have will be something to do with mathematics.

The term "Idiot Savant" is actually outdated, since the term "idiot" itself is no longer a medical term. The term "savant syndrome" is more correct. It refers to someone—usually someone with a developmental disorder, often autism—who has one or sometimes more extremely high abilities that are disproportionate to his normal level of functioning. The other wiki has more information.

This trope is about people with an actual mental disorder. If the character is simply ditzy, then that's Genius Ditz.

Examples:

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Anime & Manga

Komugi of Hunter × Hunter can really not do anything at all, due in part to her blindness, general Self-Deprecation and the belief hammered into her by her parents that if she ever lost a game of Gungi ever, she is nothing more than trash... but she is quite literally undefeatable at Gungi to make up for this, being able to lead and control a game of it hundreds of moves ahead of her opponents and always increasing her skill with every match, making anyone she fights completely unable to catch up to her no matter how hard they try.

Mashiro Shiina in The Pet Girl of Sakurasou is an internationally famous painter by high school age, but she clearly has zero skills in anything else—she doesn't know how to put on her shoes, or that she has to pay for items in a store, etc, etc, together with a long list of oddities (like No Social Skills or having a flat affect) to the point that the speculation of her being autistic has became a Word of Dante.

Films — Live-Action

Rain Man: Raymond suffers from extreme autism and is usually restricted to a home. However, he has savant abilities with numbers, allowing him to count with lightning speed, do complex calculations in his head, and count cards.

The main character of Forrest Gump, particularly with things that require speed (running, loading weapons, ping-pong, etc) in combination with an extreme, single-minded concentration. This was actually toned down from the book.

The first Cube has Kazan, a severely mentally handicapped man who, to the rest of the people trapped in the Cube, is a nuisance at first, constantly banging his head against the walls, making weird honking noises and babbling about gumdrops. But later on it's discovered that he could find the prime factors of huge numbers in his head, and he ends up as their savior.

In Cube Zero, this is creepily hinted at with Wynn as having become this at the end. He's lobotomized, thrown back in the Cube, and found by some other prisoners, mirroring Kazan's introduction. The last shot of the film settles on his tapping fingers as a computer interface is softly heard in the background...

White from Tekkonkinkreet plays this trope straight. He saves Black's life, perhaps more then once, and is eerily perceptive about some of the people around him: at the same time, he can't tie his own shoes or dress himself without help.

Mercury Rising centers on an autistic child who can intuitively decipher top military ciphers, and a police officer who can convey character with a silent telegenic stare.

The film The Wizard features a young autistic boy who is fantastic at video games.

Literature

Ton Ton in the novel House of the Scorpion has trouble speaking and is regarded as a dumb ox by his peers, but can work almost any machine.

One book in the Baby Sitters Club series had Kristy sitting for a girl with autism. The girl, Susan, was mostly non-verbal and was generally not able to take care of herself, but was a brilliant piano player AND could do the "identify the day of the week" calendar trick.

Adus of The Elenium is an evilexample. A mentally handicapped, barbaric savage, Adus can barely speak, doesn't bathe, is almost illiterate, and frequently eats raw meat. He's also a combat savant, with a talent not only for physical fighting, but small unit tactics. Give him an axe and a manageable number of troops, and no one is more dangerous.

Steapa Snotor of The Saxon Stories is so stupid that he's nicknamed "The Clever", and is reckoned by the hero to be completely lacking in imagination. He is uneducated, illiterate, and can spend hours wrapping his head around the simplest of situations. The exception to this is on the battlefield, where he is both a terrifying warrior and an extremely capable commander. Alfred the Great, a man who admires only learning and otherwise only trusts literate and intelligent men, promotes Steapa to command of his personal guard, which frequently means effective command of his entire army.

In My Sweet Audrina by V. C. Andrews, Audrina's sister Sylvia is born with a severe intellectual disability which means that she can't speak, walk properly, close her mouth or control her bodily functions. Despite this, she is presented as preternaturally wise and has powers of telepathy which allow her to communicate with Audrina, ultimately leading to her coming up with a very clever solution to save the comatose Audrina from Vera, who was about to switch off Audrina's life support machine.

The Tommyknockers: smart enough to create DC-powered anti-gravity, dumb enough to never think of buying an AC-DC converter so they don't have to power it with batteries.

Live Action TV

On Boy Meets World there's a throwaway gag in one episode in which Shawn refers to himself as one after revealing he can speak French. This was during the period in the show where he got really dumb, which didn't stick as a character trait so this is never brought up again.

In The Lost Experience, the Hanso Foundation uses autistic savants as human computers in the basement of a mental hospital.

"Coach" Ernie Pantusso in Cheers is implied to be The Ditz due to head injuries, but he's also a geography expert, even making up mnemonic songs about countries.

One episode of NUMB3RS featured a very Rain-Man-like autistic savant character, right down to being able to count spilled toothpicks quickly and accurately. He was employed by a courier company to help track packages with missing or damaged bar-codes, and when he was off work, they had to bring in at least ten people to do his job.

Rom, a recurring Ferengi on Star Trek: Deep Space Nine, goes far beyond basic idiocy. Somehow he is an extremely capable engineer and at one point, even able to spout technobabble on the level of Spock and Data. His skills seem to have been honed by the Ferengi's natural skill with mathematics and the years working as Quark's personal repairman for anything and everything.

One episode of Law & Order had a suspect that could do quantum physics but couldn't remember a grocery list.

Paisley from A.N.T. Farm can't read or remember her name, but she can build a functioning helicopter complete with rope ladder out of balloons, which she can also make into a pony complete with functioning internal organs.

Patrick, House's patient of the week played by Dave Matthews in season 3's Half-Wit was this due to a car accident he was in as a kid. He couldn't even button his own shirt, yet he was a brilliant pianist despite never having practiced it.

The Tick. While he was always a ditz, the live-action television series plays up his ditzy traits so much that he seems barely capable of functioning in society. Notably, he doesn't even know his own real name or where he's from, he can't recognize other superheroes when they're out of costume (the other characters might have trouble at first, but The Tick never picks up on it even when it's explained to him), and at first doesn't even know what death is. When it comes to heroics, however, he's a genius: he's an expert fighter and shows a decent amount of philosophical thought. Notably, when he did discover that everyone (even potatoes) can die, he spent about 30 seconds thinking about it before giving Arthur a motivating speech that actually makes sense.

Melvin Potter in Daredevil is childlike and unintelligent, but a savant when it comes to making armored combat suits. He takes an authoritative tone when he's in his element, but is childlike the rest of the time.

Leland Owlsley: He's half an idiot.

Wilson Fisk: It's the other half that counts.

This element is dropped from him in season 2 as he's much more mature, no doubt the result of Fisk no longer being there to pressure him.

Murdoch Mysteries: "Dead End Street" features a woman who is almost mute and regarded as an 'imbecile' (in the parlance of the era), but has total recall and a phenomenal ability in making models. When she makes a scale model of her street with a tiny figure holding a rifle at the window, Murdoch realises she has witnessed a murder and reconstructed it in the model. However, because she cannot distinguish faces, Murdoch has no way of identifying the figure at the window and has to work out a means of communicating with her to discover what she knows.

Reese from Malcolm in the Middle. He is the dumbest person on the show, but he is genius in the kitchen and very naturally gifted

Music

The Gorillaz singer 2D may well be one of these, depending on how you look at it. A long history of head injuries and painkillers have made him pretty dense, sometimes verging on Too Dumb to Live, and yet he occasionally makes some strikingly philosophical or socially perceptive comments. At the very least he's a musical prodigy, writing and performing beautiful melodies unaided. (He's supposedly responsible for the entire album, "The Fall.")

Professional Wrestling

WWE once featured a wrestler called Eugene who was billed as an idiot savant. He was mentally and socially way below the norm, but his wrestling abilities were pretty good, including pulling off numerous classical moves from the old days of pro-wrestling.

Tabletop Games

Exalted: The Primordials are often referred to by fans and writers alike as idiot savants. This is directly enshrined in their powers; when acting within their defining themes, even the basic abilities of a Primordial are transcendentally incredible. In any area outside of their themes, they are utterly hopeless. The relevance of these matters is a bit bigger considering that they are the immense and barely comprehensible Titans who forged the world and created all mortal and godly races; since those tended to be group efforts, the Primordials were never really fully capable of grasping the consequences or possibilities of their creations. It did not work out well for them.

However, this only applies to the period from Manual of Exalted Power: Infernals to the end of second edition. Outside of that period, they're still portrayed as vast and barely comprehensible, but they're not idiot savants.

The Jokaero of Warhammer 40K are basically space Orangutans, in both appearance and intellect. Yet somehow, they are also mechanical craftsmen par-excellance.

Sandal, from the Dragon Age: Origins, is a savant when it comes to enchanting weapons. However, if you talk to him all he'll say is "Enchantment!" and "Enchantment?", probably owing to the amount of lyrium he's been exposed to. (Although the sequel reveals he is able to speak, he's just shy and inarticulate.) In the final dungeon at the end of the game you also encounter him alone, covered in blood, and surrounded by heaps of slain monster corpses. So he would appear to be a bit of a savant when it comes to combat as well.

Warden: "You're surrounded by darkspawn corpses! What happened here?"

Sandal: "Enchantment!"

Mass Effect 2 has a DLC side mission where Shepard takes on a rogue Virtual Intelligence that has melded to the mind of the project lead's autistic brother. Through a series of holograms projected from the brother's mind, Shepard learns that not only was he a math genius, but he could actually imitate the geth language so accurately that the deactivated geth would respond to it.

The Fallout 2 NPC Algernon is an idiot savant with a talent of fiddling with guns and upgrading them. The poor guy lives in New Reno, a city controlled by mobsters, and lacks the character and social acumen to hold his own there, so an unscrupulous gun shop owner hides him in his basement and makes him work for food. If the player character kills said owner, Algernon will help them with their guns for free.

One of the Luck perks in Fallout 4 is named this. It gives the player a random chance of getting hugely multiplied experience points from any action, said chance being higher the lower their intelligence is.

Web Animation

Goku from Dragon Ball Z Abridged puts virtually all his brainpower towards fighting. Hence why he's good at fighting and fighting-related analysis, and near-brain dead stupid at everything else. He is implied to have suffered brain damage as a child.

Web Comics

Ethan in Ctrl+Alt+Del, emphasis on idiot (he thinks the best way to deal with paperwork is to Kill It with Fire). He claims he's intelligent for only a few seconds a day and created two intelligent robots without fully understanding how, since the only plans he could find were scribbled in sauce on a restaurant menu. When one of them needs to be fixed, he gives a very technical explanation why he can't do it. The next strip shows him wearing all his winter clothes instead of, y'know, putting them on a chair or something because he needed the storage space.

Western Animation

The Simpsons: The male Simpsons can be classified as this at their worst. While being born smart, a gene causes that to deteoriate. However, both Homer and Bart have displayed excellent skills in multiple languages (Homer once speaking penguin). To say nothing of musical talent. Bart showed great skill with the drums, and Homer has repeatedly shown himself to be capable at anything - literally anything - connected to music. Two hit bands, several successful songs both with those bands and independently - he even shows incredible skill as a music manager, for both Lurleen Lumpkin and Lisa.

It's never directly addressed, but in G.I. Joe, Metalhead, Destro's lackey, is one of the dumbest characters around, but can apparently do complex artillery calculations for his rockets in his head. He also has one of the best accuracy rates of any Joe or Cobra, mostly because he only fires at vehicles.

Subverted when Charmcaster calls Gwen Tennyson this exact name in Ben 10: Ultimate Alien, referring to Gwen's incredible but self-taught magical powers. "I've trained in magic my whole life, and you just pick up my spell book and you're instantly out-magicking me. You're not a real sorceress, you're an idiot savant." Gwen is a more realistic version of a Teen Genius and likely knows what an idiot savant is. Charmcaster was insulting her out of envy for her natural talent (due to being part anodite, a being of pure mana, the substance of magic).

Lampshaded in an episode of Family Guy when the gang play laser tag and talk about how Peter is amazing at it, all while he's rolling on the ceiling and other crazy stunts.

Joe: Alright, keep an eye out for Peter, he's oddly competitive at this stuff. Quagmire: Yeah, he seems to be weirdly physically fit at this place. Brian: Yeah, it's almost like he has unrealistic abilities when we play laser tag.

Fry in Futurama is this; it turns out is a naturally-talented composer, at least for the Holophoner, though he lacks the dexterity to actually play it properly. In The Devil's Hands Are Idle Playthings, he switches his hands with the Robot Devil's and becomes an overnight musical sensation. This may be a result of his strange mind due to being his own grandfather.

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